Summary: On Continuous Normalization
Klaus Aehlig ? and Felix Joachimski
faehligjjoachskig@mathematik.uni-muenchen.de
Mathematisches Institut, Ludwig-Maximilians-Universitat Munchen
Theresienstrasse 39, 80333 Munchen, Germany
Abstract. This work aims at explaining the syntactical properties of
continuous normalization, as introduced in proof theory by Mints, and
further studied by Ruckert, Buchholz and Schwichtenberg.
In an extension of the untyped coinductive -calculus by void construc-
tors (so-called repetition rules), a primitive recursive normalization func-
tion is dened. Compared with other formulations of continuous normal-
ization, this denition is much simpler and therefore suitable for analysis
in a coalgebraic setting. It is shown to be continuous w.r.t. the natural
topology on non-wellfounded terms with the identity as modulus of con-
tinuity. The number of repetition rules is locally related to the number
of -reductions necessary to reach the normal form (as represented by
the Bohm tree) and the number of applications appearing in this normal
form.
Introduction
Continuous normalization has been introduced by Mints [Min78,KMS75] in order